Abstract
Dense operators for preconditioning sparse linear systems have traditionally been considered infeasible due to their excessive computational and memory requirements. With the emergence of techniques such as block low-rank approximations and hierarchical multipole approximations, the cost of computing and storing these preconditioners has reduced dramatically. In our prior work [15], we have demonstrated the use of multipole-based techniques as effective parallel preconditioners for sparse linear systems. At one extreme, multipole-based preconditioners behave as dense (bounded interaction) matrices (multipole degree 0), while at the other extreme, they are represented entirely as series expansions. In this paper, we show that: (i) merely truncating the kernel of the integral operator generating the preconditioner leads to poor convergence properties; (ii) far-field interactions, in the form of multipoles, are critical for rapid convergence; (iii) the importance and required accuracy of far-field interactions varies with the complexity of the problem; and (iv) the preconditioner resulting from a judicious mix of near and far-field interactions yields excellent convergence and parallelization properties. Our experimental results are illustrated on the Poisson problem and the generalized Stokes problem arising in incompressible fluid flow simulations.
